The problem I had to solve was to prove exp(log(x)=x
So I don't know actually yet, according to the exercise, that this is true
But I know the derivatives of exp and log, and I know there values in 0 and 1. So I use this:

Now if i take the derivative of exp(log(x)), I get exp(log(x))*1/x
This must be equal to 1 then. Then exp(log(x)) must be x, because then
exp(log(x))/x=1, but I can't use this, because this is what I have to prove
Can't I use something else?

The problem I had to solve was to prove exp(log(x)=x
So I don't know actually yet, according to the exercise, that this is true
But I know the derivatives of exp and log, and I know there values in 0 and 1. So I use this:

Now if i take the derivative of exp(log(x)), I get exp(log(x))*1/x
This must be equal to 1 then. Then exp(log(x)) must be x, because then
exp(log(x))/x=1, but I can't use this, because this is what I have to prove
Can't I use something else?

Thanks

what does the relationship in question remind you of? it reminds me of the relationship between inverse functions. recall that, a function is invertible if there exists a function such that:

so answering your question amounts to proving that the log is the inverse of the exponential function. can you do that?

The problem I had to solve was to prove exp(log(x)=x
So I don't know actually yet, according to the exercise, that this is true
But I know the derivatives of exp and log, and I know there values in 0 and 1. So I use this:

Now if i take the derivative of exp(log(x)), I get exp(log(x))*1/x
This must be equal to 1 then. Then exp(log(x)) must be x, because then
exp(log(x))/x=1, but I can't use this, because this is what I have to prove
Can't I use something else?

I would like to know more about the background of this question. There is a widely use calculus textbook by Salas and Hille that asks this very question. That text defines the logarithm function as: .
Further, they define e as the number such that .

From these, the authors develop the usual properties of both Log(x) and exp(x).